shamirs

Minimal pure-Python implementation of Shamir's secret sharing scheme.

Purpose

This library provides functions and data structures for computing secret shares given an integer input value and for reassembling an integer from its corresponding secret shares via Lagrange interpolation over finite fields (according to Shamir's secret sharing scheme). The built-in secrets.randbelow function is used to generate random coefficients. The lagrange library is used for Lagrange interpolation.

Installation and Usage

This library is available as a package on PyPI:

python -m pip install shamirs

The library can be imported in the usual manner:

import shamirs

Examples

The library provides the function shares for transforming a nonnegative integer plaintext into a number of secret shares and the function interpolate for reassembling those shares back into the plaintext they represent:

>>> ss = shamirs.shares(123, quantity=3)
>>> len(ss)
3
>>> shamirs.interpolate(ss)
123
>>> ss = shamirs.shares(456, quantity=20, modulus=15485867, threshold=10)
>>> shamirs.interpolate(ss[5:15], threshold=10)
456

Individual secret shares are represented using the share class. This class is derived from the tuple type and can have either two integer components (the share index and the share value that together determine the coordinates of a point on a polynomial curve) or three integer components (also including the modulus). One advantage of the two-component variant is that the memory footprint of share objects is reduced. These components can be accessed either directly via their indices or via named attributes:

>>> s = shamirs.share(1, 2, 3)
>>> s.index
1
>>> s.value
2
>>> s.modulus
3
>>> [s[0], s[1], s[2]]
[1, 2, 3]
>>> int(s) # Share value.
2

The shares function accepts an optional boolean argument compact, making it possible to create secret shares that do not include the modulus component:

>>> shamirs.shares(123, quantity=3, modulus=1009, compact=True)
[share(1, 649), share(2, 778), share(3, 510)]

It is also possible to extend a two-component share object with a modulus using the built-in modulus operator (thanks to the __mod__ method):

>>> s = shamirs.share(1, 2)
>>> s % 3
share(1, 2, 3)
>>> t = shamirs.share(1, 2)
>>> t %= 3
>>> t
share(1, 2, 3)

Addition of share objects and multiplication of share objects by a scalar are both supported via special methods such as __add__ and __mul__ that correspond to Python's built-in addition and multiplication operators:

>>> (r, s, t) = shamirs.shares(123, 3)
>>> (u, v, w) = shamirs.shares(456, 3)
>>> shamirs.interpolate([r + u, s + v, t + w])
579
>>> (r, s, t) = shamirs.shares(123, 3)
>>> r *= 2
>>> s *= 2
>>> t *= 2
>>> shamirs.interpolate([r, s, t])
246

When creating secret shares for a given plaintext, the modulus can be specified explicitly or omitted. When the modulus is omitted, the default is the 128-bit modulus (2 ** 127) - 1:

>>> (r, s, t) = shamirs.shares(123, 3)
>>> r.modulus == (2 ** 127) - 1
True
>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009)
>>> r.modulus
1009

When using interpolate to reconstruct a plaintext from compact share objects, the modulus must be specified explicitly:

>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009, compact=True)
>>> shamirs.interpolate([r, s, t])
Traceback (most recent call last):
  ...
ValueError: modulus is not found in share objects and is not provided as an argument
>>> shamirs.interpolate([r, s, t], modulus=1009)
123

The reconstruction threshold can also be specified explicitly or omitted. When it is omitted, the default threshold is the number of secret shares requested:

>>> (r, s, t) = shamirs.shares(123, 3)
>>> shamirs.interpolate([r, s, t]) # Three shares (at threshold).
123
>>> shamirs.interpolate([r, s]) # Two shares (below threshold).
119174221476707020724653887077758571505
>>> (r, s, t) = shamirs.shares(123, 3, threshold=2)
>>> shamirs.interpolate([r, s]) # Two shares (at threshold).
123
>>> shamirs.interpolate([s, t]) # Two shares (at threshold).
123
>>> shamirs.interpolate([r, t]) # Two shares (at threshold).
123

The threshold argument is never required when invoking interpolate, but it can reduce the number of arithmetic operations performed when reconstructing a plaintext:

>>> ss = shamirs.shares(123, 256, threshold=2)
>>> shamirs.interpolate(ss) # Slower.
123
>>> shamirs.interpolate(ss, threshold=2) # Faster.
123

To facilitate rapid prototyping and assembly of concise tests, the add and mul helper functions are provided for performing addition and scalar multiplication operations involving collections of shares:

>>> ss = shamirs.shares(123, 3)
>>> ts = shamirs.shares(456, 3)
>>> shamirs.interpolate(shamirs.add(ss, ts))
579
>>> shamirs.interpolate(shamirs.mul(ss, 2))
246

These methods can also be used instead of built-in arithmetic operators when working with share objects that have no modulus component:

>>> (r, s, t) = shamirs.shares(123, 3, modulus=1009, compact=True)
>>> (u, v, w) = shamirs.shares(456, 3, modulus=1009, compact=True)
>>> shamirs.interpolate(
...     [
...         shamirs.add(r, u, modulus=1009),
...         shamirs.add(s, v, modulus=1009),
...         shamirs.add(t, w, modulus=1009)
...     ],
...     modulus=1009
... )
579
>>> shamirs.interpolate(
...     [
...         shamirs.mul(r, 2, modulus=1009),
...         shamirs.mul(s, 2, modulus=1009),
...         shamirs.mul(t, 2, modulus=1009)
...     ],
...     modulus=1009
... )
246

In addition, conversion methods for bytes-like objects and Base64 strings are included to support encoding and decoding of share objects:

>>> shamirs.share.from_base64('AQAAAAIAAADkAPED').to_bytes().hex()
'0100000002000000e400f103'
>>> [s.to_base64() for s in shamirs.shares(123, 3, 1009)]
['AQAAAAIAAADkAPED', 'AgAAAAIAAABRAfED', 'AwAAAAIAAADCAfED']

Development

All installation and development dependencies are fully specified in pyproject.toml. The project.optional-dependencies object is used to specify optional requirements for various development tasks. This makes it possible to specify additional options (such as docs, lint, and so on) when performing installation using pip:

python -m pip install ".[docs,lint]"

Documentation

The documentation can be generated automatically from the source files using Sphinx:

python -m pip install ".[docs]"
cd docs
sphinx-apidoc -f -E --templatedir=_templates -o _source .. && make html

Testing and Conventions

All unit tests are executed and their coverage is measured when using pytest (see the pyproject.toml file for configuration details):

python -m pip install ".[test]"
python -m pytest

Alternatively, all unit tests are included in the module itself and can be executed using doctest:

python src/shamirs/shamirs.py -v

Style conventions are enforced using Pylint:

python -m pip install ".[lint]"
python -m pylint src/shamirs

Contributions

In order to contribute to the source code, open an issue or submit a pull request on the GitHub page for this library.

Versioning

Beginning with version 1.0.0, the version number format for this library and the changes to the library associated with version number increments conform with Semantic Versioning 2.0.0.

Publishing

This library can be published as a package on PyPI via the GitHub Actions workflow found in .github/workflows/build-publish-sign-release.yml that follows the recommendations found in the Python Packaging User Guide.

Ensure that the correct version number appears in pyproject.toml, and that any links in this README document to the Read the Docs documentation of this package (or its dependencies) have appropriate version numbers. Also ensure that the Read the Docs project for this library has an automation rule that activates and sets as the default all tagged versions.

To publish the package, create and push a tag for the version being published (replacing ?.?.? with the version number):

git tag ?.?.?
git push origin ?.?.?